# L11a269

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a269 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^2 t(2)^5+t(1)^3 t(2)^4-4 t(1)^2 t(2)^4+3 t(1) t(2)^4-2 t(1)^3 t(2)^3+8 t(1)^2 t(2)^3-7 t(1) t(2)^3+2 t(2)^3+2 t(1)^3 t(2)^2-7 t(1)^2 t(2)^2+8 t(1) t(2)^2-2 t(2)^2+3 t(1)^2 t(2)-4 t(1) t(2)+t(2)+t(1)}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $q^{3/2}-3 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{15}{q^{5/2}}-\frac{18}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^7 z^5+3 a^7 z^3+4 a^7 z+2 a^7 z^{-1} -a^5 z^7-4 a^5 z^5-7 a^5 z^3-7 a^5 z-3 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-6 a^3 z^3-3 a^3 z+a^3 z^{-1} +a z^5+3 a z^3+2 a z$ (db) Kauffman polynomial $a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-4 a^{10} z^4+a^{10} z^2+6 a^9 z^7-11 a^9 z^5+10 a^9 z^3-5 a^9 z+6 a^8 z^8-6 a^8 z^6-a^8 z^4+2 a^8 z^2+4 a^7 z^9-3 a^7 z^5-4 a^7 z^3+6 a^7 z-2 a^7 z^{-1} +a^6 z^{10}+10 a^6 z^8-21 a^6 z^6+17 a^6 z^4-9 a^6 z^2+3 a^6+7 a^5 z^9-10 a^5 z^7+9 a^5 z^5-15 a^5 z^3+12 a^5 z-3 a^5 z^{-1} +a^4 z^{10}+8 a^4 z^8-22 a^4 z^6+22 a^4 z^4-13 a^4 z^2+3 a^4+3 a^3 z^9-a^3 z^7-9 a^3 z^5+9 a^3 z^3-2 a^3 z-a^3 z^{-1} +4 a^2 z^8-9 a^2 z^6+5 a^2 z^4-a^2 z^2+a^2+3 a z^7-9 a z^5+8 a z^3-2 a z+z^6-3 z^4+2 z^2$ (db)

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-8-7-6-5-4-3-2-10123χ
4           1-1
2          2 2
0         41 -3
-2        72  5
-4       95   -4
-6      96    3
-8     99     0
-10    79      -2
-12   59       4
-14  37        -4
-16  5         5
-1813          -2
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.